On principal element lattices (Q2663184)

The authors study principal element lattices, Prüfer lattices and \(Q\)-lattices. A necessary and sufficient condition is given for a principally generated \(C\)-lattice to be a finite direct product of proper Dedekind domains. A multiplicative lattice \(L\) is said to satisfy the condition (**) (respectively, (***)) if it has a multiplicatively closed set \(S\) of (not necessarily principal) elements which generates \(L\) under joins such that each element of \(S\) is a finite product of primary elements (respectively, a finite meet of prime power elements). Principal element lattices are characterized in several ways. The following results are proved. Theorem. Suppose \(L\) satisfies condition (**). Then the following statements on \(L\) are equivalent: \begin{itemize} \item[(1)] \(L\) is a principal element lattice. \item[(2)] Every maximal element is locally principal. \item[(3)] Every maximal element is a compact l-prime. \item[(4)] \(L\) satisfies the union condition on primes and every primary element is a power of its radical. \item[(5)] Any two incomparable primary elements are comaximal and every idempotent maximal is unbranched. \end{itemize} Theorem. Suppose \(L\) satisfies condition (***) and the union condition on primes. Then the following statements on \(L\) are equivalent: \begin{itemize} \item[(1)] \(L\) is a principal element lattice. \item[(2)] Every semiprimary element is primary. \item[(3)] Every primary element is a power of its radical. \item[(4)] \(L\) is an \(\alpha\)-lattice. \item[(5)] Every non-minimal prime element is weak meet principal. \item[(6)] \(L\) is a Prüfer lattice. \end{itemize}